A new model to predict soil thermal conductivity

Thermal conductivity is a basic parameter of soil heat transferring, playing an important role in many fields including groundwater withdrawal, ground source heat pump, and heat storage in soils. However, it usually requires a lot of time and efforts to obtain soil thermal conductivity. To conveniently obtain accurate soil thermal conductivity, a new model describes the relationship between soil thermal conductivity (λ) and degree of saturation (Sr) was proposed in this study. Dry soil thermal conductivity (λdry) and saturated soil thermal conductivity (λsat) were described using a linear expression and a geometric mean model, respectively. A quadratic function with one constant was added to calculate λ beyond the lower λdry and upper λsat limit conditions. The proposed model is compared with five other frequently used models and measured data for 51 soil samples ranging from sand to silty clay loam. Results show that the proposed model match the measured data well. The proposed model can be used to determine soil thermal conductivity of a variety of soil textures over a wide range of water content.


Proposed model. Previous studies show that values calculated by the
model is much smaller than the measured values [42][43][44]51 . Enhancing the Campbell (1985) model calculations requires calibration for different soils to obtain the model parameters which increases the complexity of the model. To improve the agreement between calculated and measured data, we modified the Campbell (1985) model by replacing θ with S r and changing the exponential term. Upon examining the variation curve of soil thermal conductivity, it becomes evident that there is a linear increase in thermal conductivity with increasing water content within the range of 0-0.2 m 3 /m 3 (Fig. 1). The propose model is split into three terms. The first two terms of the model consist of a linear function related to the saturation P+QS R r . Since the thermal conductivity of different soil types increases with different rates. A parameter R was used to control the slope of the calculated λ plot at lower water content. Besides, a parameter S was introduced to control the calculations beyond the lower (λ dry ) and upper (λ sat ) limit conditions in the third item of the proposed model (S exp[S r (1 − S r )]) . The modified model is applicable for different types of soil, which can be described as following: www.nature.com/scientificreports/ where P, Q, and R are related to physical properties of the soil (porosity, composition, and texture); S is related to S r ; S r is soil saturation, % and c is a constant; λ dry is dry soil thermal conductivity, W/m °C and is saturated soil thermal conductivity, W/m °C. Both the Tien (2005) model 47 and the Li (2008) model 42 are modifications of the Campbell model. However, the two models only adjust the coefficients of the Campbell model based on their measured data, without incorporating the variations in thermal conductivity. In contrast, the Johansen (1975) and Lu et al. (2007) models propose a linear relationship to calculate λ between λ sat , calculated using the geometric mean model 37 , and λ dry with K e . Despite the significant improvement in model accuracy, the complex logarithmic or exponential formulations increase the computational difficulty of the two models. Many models have been developed to predict saturated soil thermal conductivity (λ sat ) and soil solids thermal conductivity (λ s ) 25, 52-55 . He et al. 30 pointed out that the geometric mean model proposed by Woodside and Messmer 37 is one of the most frequently used models. The model was described as following: where n is soil porosity, w (0.594 W/m °C) is the thermal conductivity of water, 0 is taken as 2.0 W/m °C for soils with q > 0.2, and 3.0 W/m °C for soils with q ≤ 0.2, q is the quartz content in the soil, q (7.7 W/m °C) is the thermal conductivity of quartz.
The Campbell (1985) model is used as a reference model. Two modified Campbell models by Tien et al. 47 and Li et al. 42 , the Johansen (1975) model, and the Lu et al. (2007) model were also used as reference models to compare with the proposed model. Soil samples. The TEMPOS Thermal Properties Analyzer (TTPA) from METER Group (U.S.) which can read four different sensors was used to measure λ. TR-3 sensor (100 mm in length, 2.4 mm in diameter) was used to obtain λ within 1 min after inserting the sensor into soil. The values of θ were measured by Stevens Hydra probe II temperature and humidity probe.
We collected 8 soil samples from field. The 8 soil samples have different textures including sand, sandy clay loam, silty clay, loam, sandy loam, silt loam, and silt clay loam. Soil samples were air dried, ground, and sieved through a vibrating screen. Then the soil samples with different particle sizes were made according to the international standard for soil texture 56 . To reduce the impact of environment on the results, laboratory temperature was controlled at constant (20 °C) by air conditioner while measuring λ. Different soil water contents were obtained by adding a certain amount of water to the soil sample and thoroughly mixing the water and soil. Firstly, the 8 soil samples with different initial gravimetric water content of 0% (dry soil), 3%, 5%, 8%, 10%, 13%, 15%, 18%, and 20% were sealed and kept for 6 h, respectively. Secondly, the 8 soil samples with different gravimetric water contents were packed into sealed columns of Φ105 × 110 mm (diameter × height) and placed at least 12 h to ensure uniform distribution of water in soil. The porosity of soils was controlled by dry density which was govern through the height and mass of the soil in sealed columns. After setting for 12 h, λ was measured 3 times for each soil sample by a TR-3 sensor.
Measured datasets in the literature were also collected to verify the proposed model. The Tarnawski et al. 57 dataset consisting of 39 soils from nine Canadian provinces and the Lu et al. 29 dataset with 12 soils (10 from China, 2 from U.S.) were used. Detailed information on the texture and particle-size of all the 59 soils are listed in Table 1 is the average of measured soil thermal conductivity. The value of RMSE represents the efficiency of this model. The lower the value, the better the reliability of the model. PBIAS indicates the total amount of disparities between the calculated and measured values. PBIAS lower than 0.10 is desired, between 0.10 and 0.15 is reasonable, between 0.15 and 0.25 is satisfactory, and above 0.25 is not satisfied 58 . The value of NSE is less than 1, which is perfect above 0.75, good between 0.65 and 0.75, satisfactory between 0.5 and 0.65, and not satisfied below 0.5 59 .

Results and discussion
Relationship between soil thermal conductivity (λ) and volumetric water content (θ). Figure 1 presents measured λ as a function of θ for the 8 soil samples with different textures. The influence of soil texture on the shape of λ curve has three distinct characteristics. First, the 8 soil samples with different textures have similar λ when the soil is dry, namely texture has little influence on λ dry . Second, sand (soils 2 and 3) has higher λ values than other soils and the greatest λ value appears in the soil which has the highest quartz content (q). Besides, at lower volumetric water contents, values of λ increased more gradually on fine-texture soils (soils 6, 11, 12, and 34); while λ values of sand (soils 2 and 3) and clay (soils 8 and 39) exhibit the fastest and the lowest rate of growth, respectively. The θ at which λ sharply increases is greater for clay than those for sand and finetextured soils. This can be explained by the fact that clay has larger surface areas and more water is required before water bridges forms between soil solid particles 29,60 .
The influence of θ on λ at room temperature can be explained by the process of substituting water for air between soil particles. Among solid, liquid, and air phases, the air phase has the lowest thermal conductivity (0.026 W/m °C), where water thermal conductivity (0.594 W/m °C) is 22 times greater than it. By analyzing the published datasets, four domains of soil water content were subdivided by Tarnawski and Gori 61 , i.e. residual, transitory meniscus, micro/macro porous capillary, and superfluous. At lower water content (residual water domain), water molecules adhere tightly to the surface of soil particles and the thickness of water film increases slowly with the increase of θ. Consequently, the soil thermal conductivity increases gradually but not significantly. With the further increase of θ (transitory meniscus, micro/macro porous capillary water domain), water bridges are constantly formed between soil particles, which leads to the increase of the contact area between soil particles and the rapid increase of soil thermal conductivity. This process does not stop until the air in the solid particle is completely replaced by water and the thermal conductivity is no longer increased (superfluous water domain).

Determination of parameters.
At room temperature, most of the previous studies considered that porosity and quartz content are the main influencing factors of soil thermal conductivity 38, 62-64 . Chen 25 also pointed out that the leading paths for thermal conduction between solid particles in a dry state are confined to particles contact points. In this paper, n is considered as the main factor affecting λ dry . Figure 2 shows the relationship between λ dry and n. Clearly, λ dry decreases linearly with n. Therefore, a simple linear formula similar to that of Lu et al. 29 , was developed to calculate λ dry . By fitting Eq. (7) to the data in Fig. 2, the calculated values of a and b were 0.51 and − 0.6, respectively. Figure 3 shows that the geometric mean model proposed by Woodside and Messmer 37 has a good performance in calculating the soil thermal conductivity (λ sat ). He et al. 30 verified the applicability of the geometric mean model, which also indicated a good performance. So, λ sat was determined by Eqs. (10) and Eq. (11).
Tarnawski and Gori 61 found that λ remained constant from dryness to a certain critical value of water content and it varied with soil texture. Clay and fine-textured soils have larger surface areas, so λ values increased more gently at low saturation while the λ values of sand had the fastest rate of growth. Johansen 38 and Lu et al. 29 also A 1stOpt program 65 was used to find the optimized value of S using the Levenberg-Marquardt algorithm. Figure 3 presents the values of S as a function of S r for the 8 soil samples. By fitting Eq. (9) to the optimized value of S in Fig. 4, the fitted value of constant c was 1.5.
Therefore, the proposed model can be described as followed: Model comparison for the eight typical soil samples. Figure 5 presents the comparison between the measured and calculated λ for the 8 typical soil samples. RMSE, PBIAS, and NSE of different models are listed in Table 2. The quartz contents of the 8 soils were not measured, so the measured λ sat was used to estimate λ of the soils. Models of Campbell 41 , Tien et al. 47 , and Li et al. 42 produced large deviations in the entire water   Table 2, the new model has the best performance on sand and the RMSE, PBIAS and NSE are 0.04 W/m °C, − 0.03 and 0.99, respectively. Although we did not obtain the exact quartz content, actual λ and quartz content measurements were listed in the paper 57 . Figure 5e-h show that the accuracy of the proposed model can be high.

Model evaluation and comparison using data in the literature.
To further evaluate the accuracy of the proposed model, λ of soils with different textures and from different regions were used. Published datasets from Tarnawski et al. 57 and Lu et al. 29 and measured data in this study (soils 1, 4, 5, and 7) were used to evaluate the six models. Lu et al. did not measure the quartz content, either. So, it was assumed that the quartz content equals to the sand content 29 . The results are shown in Fig. 6a-f and RMSE, PBIAS and NSE of the 6 models are listed in Table 3 In the process of model validation, we found that the Campbell (1985) model (RMSE/PBIAS/ NSE = 0.28/0.10/0.77) was greatly affected by soil bulk density, and it could better predict λ within the bulk density range of 1.2-1.4 g cm −3 . As shown in Fig. 6a, most of the symbols in the model are located below the 1:1 line, implying that the calculated values are smaller than the measured value, which is consistent with Zhao et al. 51 (Fig. 6d). The Lu et al.   (Table 3).
In general, the normalized empirical model (Johansen (1975) 2007) model, both models demonstrate similar accuracy, but they address the issue of varying growth rates of thermal conductivity at low water content stages for different soil textures differently. Lu et al. 29 employs a complex exponential equation to tackle this problem, whereas the new model resolves it by introducing a single parameter, R, which is derived from extensive analysis and synthesis of experimental data. It is widely recognized that thermal conductivity measurements are primarily conducted in field settings, where simpler models prove more advantageous for practical field work. Additionally, the new model achieves high accuracy and has been developed and verified using a diverse range of soil textures. The proposed model accurately estimates λ across the entire water content range.

Conclusions
A new model has been developed to calculate soil thermal conductivity based on water content, utilizing only a small set of simple parameters including quartz content and porosity. The new model demonstrates excellent agreement with the measurement data across the entire water content range of 8 soil samples. Moreover, it exhibits strong consistency with soils of various textures, as documented in the literature. The new model creatively solves the problem of inaccurate calculation of soil thermal conductivity in the low water content stage of existing models. These findings highlight the robustness and versatility of the proposed model, making it a valuable tool for accurately estimating the thermal conductivity of diverse soil types. The new model can be applied to distributed hydrological models in the future and can also provide approximate soil thermophysical property parameters for the construction of ground source heat pump systems. While the new model successfully addresses the challenge of accurately calculating λ for different soil textures at low water content, it is worth noting that the current introduction of empirical parameters (e.g., parameter R) lacks direct practical significance. To further improve the applicability of model, future research endeavors should focus on establishing meaningful relationships between these empirical parameters and essential soil physical properties, such as bulk density and porosity. By providing practical interpretations for these parameters, the model can be refined, expanding its applicability and contributing to a more comprehensive understanding of soil behavior. Ultimately, these efforts would enhance the accuracy of thermal conductivity calculations. www.nature.com/scientificreports/ www.nature.com/scientificreports/

Data availability
All data generated or analysed during this study are included in this published article and its appendix files.